# Properties

 Label 1077.a Conductor $1077$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

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## Genus 2 curves in isogeny class 1077.a

Label Equation
1077.a.1077.2 $$y^2 + (x^3 + 1)y = x^4 + x^3 + 2x^2 + x$$
1077.a.1077.1 $$y^2 + (x^3 + 1)y = 5x^5 + 34x^4 + 80x^3 - x^2 - 90x + 32$$

## L-function data

Analytic rank:$$1$$
Mordell-Weil rank:$$1$$

Bad L-factors:
Prime L-Factor
$$3$$$$( 1 + T )( 1 + 2 T + 3 T^{2} )$$
$$359$$$$( 1 - T )( 1 + 8 T + 359 T^{2} )$$

Good L-factors:
Prime L-Factor
$$2$$$$1 + 2 T + 3 T^{2} + 4 T^{3} + 4 T^{4}$$
$$5$$$$1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4}$$
$$7$$$$1 + T + 6 T^{2} + 7 T^{3} + 49 T^{4}$$
$$11$$$$1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}$$
$$13$$$$( 1 - 4 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$$17$$$$1 + 3 T - 4 T^{2} + 51 T^{3} + 289 T^{4}$$
$$19$$$$1 - T + 10 T^{2} - 19 T^{3} + 361 T^{4}$$
$$23$$$$1 + 4 T - 2 T^{2} + 92 T^{3} + 529 T^{4}$$
$$29$$$$( 1 + 29 T^{2} )( 1 + 6 T + 29 T^{2} )$$
$\cdots$$\cdots$

See L-function page for more information

## Sato-Tate group

$$\mathrm{ST} =$$ $\mathrm{USp}(4)$

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

## Endomorphisms of the Jacobian

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism algebra over $$\Q$$:

 $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.

More complete information on endomorphism algebras and rings can be found on the pages of the individual curves in the isogeny class.